Experimental Evidence for Resonant-Tunneling in a Luttinger-Liquid
O. M. Auslaendera , A. Yacobya , R. de Picciottob , K. W. Baldwinb , L. N. Pfeifferb , and K. W. Westb

arXiv:cond-mat/9909138v2 [cond-mat.mes-hall] 15 Sep 1999

a

Dept. of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot, 76100, Israel
b
Bell Labs, Lucent Technologies, 700 Mountain Ave., Murray Hill, NJ 07974, U.S.A.

We have measured the low temperature conductance of a one-dimensional island embedded in a
single mode quantum wire. The quantum wire is fabricated using the cleaved edge overgrowth
technique and the tunneling is through a single state of the island. Our results show that while
the resonance line shape fits the derivative of the Fermi function the intrinsic line width decreases
in a power law fashion as the temperature is reduced. This behavior agrees quantitatively with
Furusaki’s model for resonant tunneling in a Luttinger-liquid.
PACS numbers: 73.20.Dx, 73.23.Ad, 73.23.Ps, 73.50.Jt

One-dimensional (1D) electronic systems are expected
to show unique transport behavior as a consequence of
the Coulomb interaction between carriers [1]. Unlike in
two and three dimensions [2], where the Coulomb interaction affects the transport properties only perturbatively,
in 1D it completely modifies the ground state from its
well-known Fermi-liquid form and the Fermi surface is
qualitatively altered even for weak interactions. Today, it
is well established theoretically that the low temperature
transport properties of interacting 1D-electron systems
are described in terms of a Luttinger-liquid rather than
a Fermi-liquid [3,4]. The difference between a Luttingerliquid and Fermi-liquid becomes dramatic already in the
presence of a single impurity. According to Landauer’s
theory the conductance of a single channel wire with
2
2
a barrier is given by G = |t| · e2 /h, where |t| is the
transmission probability through the barrier. This result
holds even at finite temperatures, assuming the transmission probability is independent of energy, as is often
the case for barriers that are sufficiently above or below
the Fermi energy. In 1D, interactions play a crucial role
in that they form charge density correlations. These correlations, similar in nature to charge density waves [5],
are easily pinned by even the smallest barrier, resulting
in zero transmission and, hence, a vanishing conductance
at zero temperature. At finite temperatures the correlation length is finite and the conductance decreases as a
power-law of temperature, G(T ) ∝ T 2/g−2 [3,4]. Here

the wire. Finally, The strongest manifestation of interaction in the clean limit comes from tunneling experiments such as the one recently reported on single walled
carbon nanotubes [9] and those performed on the chiral
Luttinger-liquid [10].
In this work we have focused on the transport properties through confined states in a 1D wire, namely, when
a 1D island is embedded in a 1D wire. The 1D island is
formed at low densities such that the disorder potential in
the wire exceeds the Fermi energy at several points along
the wire. Resonant tunneling (RT) has previously been
studied experimentally in a chiral Luttinger-liquid when
the resonant level width was larger than the electron temperature [11,12]. However, an unequivocal verification of
the theoretical prediction has not been obtained. Theoretically, the problem of RT has been considered by Kane
and Fisher [3] and was later extended by Furusaki [13] to
include many resonant levels and the effects of Coulomb
blockade (CB). Our measurements probe the intrinsic
width, Γi , of several resonant states as the temperature
is lowered. In contrast to conventional CB theory [14],
where Γi is temperature independent, we find that Γi
decreases as a power law of temperature over our entire
temperature range (2.5K to 0.25K). The measured behavior is in quantitative agreement with the theoretical
prediction of Furusaki [13].
The 1D wires are fabricated by the CEO method (see
Fig. 1a and [6]). The electrons are confined by a 25nm
square well potential in one direction and by a triangular potential well approximately 10nm wide (binding
them to the cleaved edge). To create a 1D island within
our wire we have chosen to study 5µm long wires that
show disorder induced deviations from the conductance
plateaus (see Fig. 1b). The wire conductance is measured
as a function of its density (by negatively biasing the top
gate, see Fig. 1a) using standard lock-in techniques. A
fixed excitation voltage of 10µV is applied across the wire
and the corresponding current is measured. As the density of electrons in the wire is reduced, the 1D modes
are depopulated one by one until only a single mode
remains partially populated. Several broad resonances
appear superimposed on the ’last plateau’ with an aver-

U
g ≈ 1/ 1 + 2EF where U is the Coulomb energy between particles and EF is the Fermi energy in the wire.
Despite the vast theoretical understanding of Luttingerliquids only a handful of experiments have been interpreted using such models. For example, in clean semiconductor wires prepared by the cleaved edge overgrowth
(CEO) method [6], contrary to theory, the conductance
is suppressed from its universal value [7]. Although not
fully understood this suppression is believed to be a result of Coulomb interactions that suppress the coupling
between the reservoirs and the wire region. Other measurements done on weakly disordered wires [8] show a
weak temperature dependence of the conductance that
is attributed to the Coulomb forces between electrons in

1

age conductance of 0.45 · 2e2 /h. We attribute the broad
resonances to above barrier scattering when the Fermi
energy is higher than the disorder potential. The deviation of the conductance plateau from the universal value
has been studied in detail in [7], however, a detailed theoretical explanation is still lacking. It should be noted
that on similar but cleaner wires we have previously reported [6,7] plateau conductance values of ≈ 0.8 · 2e2 /h.
The larger deviation observed here suggests that disorder
plays an important role in the suppression of the value
of the conductance plateaus. As the density is further
reduced, the highest potential barrier in the wire crosses
the Fermi energy, the last mode is pinched off, and the
wire splits into two parts. Upon further decrease in density a second barrier crosses the Fermi energy, thereby,
forming a 1D island, and transport occurs through resonant states. Of course as the density is reduced even
further, more islands will form in the wire. However, for
transport to occur through them, at least one confined
state in each island must be concurrently aligned with the
Fermi energy. Since this condition is very unlikely, the
wire is expected to be completely pinched off when more
than one island has developed. Therefore, the sharp resonances in the sub-threshold region, being almost equally
spaced in gate voltage are attributed to CB resonances
through a single 1D island.
The conductance due to RT of a particle between two
Fermi-liquid leads is easily calculated using the Landauer
2
2
2
|t(ε)| ∂f dε, where |t(ε)| has the
formula, GF L = e
h
∂ε
Breit-Wigner line shape centered around the resonant
2
Γ2
i
energy ε0 , |t(ε)| = (ε−ε )2 +Γ2 , and f is the Fermi func0
i
tion. When kB T ≫ Γi , the case of interest here, one
2
ε0 −µ
finds that GF L = e Γi 4kπ T cosh−2 2kB T with µ the
h
B
chemical potential in the leads. The main outcome of
this analysis is the line shape of the resonance being the
derivative of the Fermi function, its full width at half
maximum equals 3.53 · kB T , and the area under the peak
(or the peak height multiplied by kB T ) is proportional to
Γi . In the conventional theory of CB [14], Γi depends on
the transmission probabilities through the individual barriers, which are independent of temperature and hence
should lead to a peak area independent of temperature.
In the case of RT from a Luttinger-liquid, the individual
transmission probabilities are suppressed as the temperature is lowered [3,4]. Therefore it is expected and has
been shown theoretically [13] that the extracted Γi should
drop to zero as Γi ∝ T 1/g−1 . The resonance line shape,
however, in the case of kB T ≫ Γi , has been shown [12,13]
to be only slightly modified by the interactions and the
change is too small to be detected experimentally. We,
therefore, deduce the electron temperature (in units of
gate voltage) from the fit to the derivative of the Fermi
function. The deduced temperature from all the resonances in Fig. 1b is the same and follows linearly the
fridge temperature. Such a fit also allows us to calibrate

the gate voltage in units of energy thereby extracting the
charging energy (distance between peaks) estimated to
be 2.2meV . Knowing the cross-section of the wire we
estimate the length of the island from the charging energy to be 100 − 200nm. It is, therefore, likely that the
1D island is connected on both sides to two 1D conductors that are several microns long. Figure 2 shows the
extracted Γi for the peaks marked in Fig. 1. It is clear
that Γi is not constant but rather drops as a power law
of temperature. The extracted values of g for the two
peaks are 0.82 (peak #1 in Fig. 1) and 0.74 (peak #2
in Fig. 1). The change in g results from the change in
density induced in the 1D wire when moving from one
peak to the next. Similar power law behavior is observed
for all measured resonances in three different wires. The
observed power law behavior is direct proof of Luttingerliquid behavior in our CEO wires.
At sufficiently high temperatures the assumption of
tunneling through a single resonant state breaks down
and we should expect an increase in the extracted Γi due
to transport through a few excited states of the 1D island [13]. The possibility of an excited state affecting the
temperature dependent conductance is of interest since
it allows a better test of Furusaki’s model. The excited
state spectrum of the 1D island is extracted from differential conductance measurements at finite source drain
voltage, Vds . Figure 3 shows a gray scale plot of the differential conductance as a function of the top gate voltage
and Vds . For peak #1 (in Fig. 1b) several excited states
can be observed. The lowest three, at Vds = −0.4meV ,
Vds = −0.7meV and Vds = −1.5meV are only very
weakly coupled (approximately 15% of the intensity of
the main peak) and would therefore contribute very little to the overall conductance. However, the fourth excited state at Vds = −1.7meV is more strongly coupled.
Since an excited state contributes to the conductance
only when 4kB T ≥ ∆E (∆E is the energy of the excited state), within our temperature range of 0.25K to
2.5K only the ground state contributes significantly and
one expects a single power law behavior as is indeed observed in Fig. 2. It should be noted that theoretically, g
can also be written in terms of the charging energy, Uc ,
and the level spacing, ∆E, as g ≈ 1/ 1 + Uc /∆E [9].
In our 1D island the ratio of Uc /∆E ≈ 5 and, hence, one
expects g ≈ 0.4. The large disagreement between the
measured g and the expected one is not understood at
this stage.
A different case is presented in Fig. 4 with a strongly
coupled excited state at Vds = −0.6meV . Hence, we expect that at temperatures above 1.2K this excited state
would contribute to the conductance. Fig. 5 shows the
temperature dependence of the extracted Γi of this CB
peak. Indeed above 1K, Γi deviates from the low temperature power law, indicating a contribution of an additional transport channel to the total conductance. At low
temperatures though, only the ground state contributes
2

to the conductance. Therefore, a fit of the low temperature data to a power law enables us to extract a g value
of 0.66 for this wire. Using this g value and the measured
energy of the excited state (-0.6meV from Fig. 4) we use
Furusaki’s model to predict the dependence of Γi over the
entire temperature range. The dashed curve in Fig. 5 is
the result of such a calculation where only the coupling
strength to the excited state has been adjusted. We see
that the temperature dependence predicted by the model
agrees quantitatively with the measured dependence, further supporting the fact that Luttinger-liquid behavior
describes the transport properties of these resonances.
In conclusion we have studied the temperature dependence of CB resonances of a 1D island embedded in an
interacting 1D wire. The observed power law behavior
of the intrinsic resonance width on temperature is direct
proof of Luttinger-liquid behavior in this system. The
measured g values range from 0.66 to 0.82 for the various
resonances studied. The measured behavior agrees quantitatively with the model of Furusaki even when excited
states of the 1D island are taken into account.
We would like to thank H. L. Stormer for the fruitful collaboration. We would also like to thank A. M.
Finkel’stein and A. Stern for helpful discussions. This
work is supported by the US-Israel BSF.

[11] F. P. Milliken, C. P. Umbach, and R. A. Webb, Solid
State Comm. 97, 309 (1996).
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(1997).
[13] A. Furusaki, Phys. Rev. B 57, 7141 (1998).
[14] L. P. Kouwenhoven et al, in Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhoven, and
G. Schon, NATO ASI series (Kluwer Academic Publishers, 1996).

FIG. 1. (a) - Top view layout of the wire and its contacting scheme. The sample is fabricated using the CEO method.
The 1D wire (thick black line) exists along the cleaved surface and overlaps a 2DEG over the entire edge. The metallic
gate depletes the 2DEG over a 5µm wide segment, thereby,
forming an isolated 1D wire that is coupled at its ends to the
overlapping 2DEG. A further increase in the top gate voltage reduces the wire density continuously to depletion. (b) Conductance of the wire as a function of the top gate voltage. Inset - A zoom-in of the conductance of the wire in the
sub-threshold region.

FIG. 2. The intrinsic line width of the resonance, Γi , vs
temperature (in units of gate voltage). Both parameters are
extracted from a fit to the derivative of the Fermi function. Γi
is seen to decrease as a power law of the temperature indicating Luttinger-liquid behavior. The dashed lines are a power
law fit to the data.
FIG. 3. A gray scale plot of the non-linear differential conductance of peak #1 (see Fig. 1b). Vds is stepped with 100µV
intervals.

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FIG. 4. A gray scale plot of the non-linear differential conductance of a resonance that has a strongly coupled state at
Vds = −0.6meV . Vds is stepped with 20µV intervals.

FIG. 5. The intrinsic line width of the resonance described
in Fig. 4 vs temperature (in units of gate voltage). The dashed
line is a fit to the data based on Furusaki’s model. g is determined from the low temperature behavior and the energy
of the excited state is determined from Fig. 4. The coupling
strength to the excited state is the only adjustable parameter
in the fit.

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